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8       TOPOLOGICAL GROUPS        37

(i) Let H be a locally closed subgroup of G. Then H is a subgroup of G,
and H is an open subgroup of H (12.2.3). Hence it is enough to prove the
second assertion. Now if H has an interior point, then by translation it
follows that every point of H is interior, and therefore H is open. Hence the
left cosets xH are also open sets, and therefore (]H is open in G (because it is a
union of cosets xH). Consequently H is closed in G.

(ii) If G is Hausdorff and H is a discrete subgroup of G, then there
exists a symmetric open neighborhood V of e such that V n H = {e}. If
x e H, we have xV n H ^ 0. Now if y e xV n H, then x e yV and the set
{y} is closed in the open set yV, because G is Hausdorff. Since yV n H = {y}
(because y e H), we must have x = y, and therefore H = H.

(12.8.8)    If G is a connected group and V is a symmetric neighborhood of e,
then G is equal to the union V00 of the sets V" for n > 0.

The set V00 is clearly symmetric, and since VmV = Vm+n it follows that
yooyoo <_ yoo Hence V is a subgroup of G. Since e is an interior point of
V00, this subgroup is both open and closed (12.8.7) and therefore is the whole
of G.

(12.8.9)    In a topological group G, the connected component K of the neutral
element (3.19) is a closed normal subgroup (called the neutral component or
identity component of G). For each x e G, the connected component ofx in G is
xK = Kx.

If a e K, the set a-1K is connected and contains e, hence K^K c K. This
shows that K is a subgroup of G. It is invariant under all automorphisms of
the topological group G, in particular under all inner automorphisms, hence
K is normal in G. Also K is closed in G (3.19). Finally, the last assertion
follows from the fact that left and right translations are homeomorphisms of
G onto G.

(12.8.10)    If Gl9 G2 are two topological groups, then the product topol-
ogy on the product group G = Gj x G2 is compatible with the   group
structure of G. For, identifying canonically the product spaces G x G and
(G! x Gx) x (O2 x G2) (12.5), the mapping

(C*i x2)9 Oi, 72))>~>C*i.);i> *2J2)

is continuous ((3.20.15) and (12.5)), and the mapping (xl9 x2)*-+(xil, x2l)
is continuous for the same reason. The group Gt x G2, endowed with the
product topology, is called the product of the topological groups G1 and G2.
If G is a commutative topological group, the mapping (x, y)i-*xy is a
continuous homomorphism of G x G into G.orff group, every discrete subgroup is closed.of the